8.3 Determinism

Hat die Zeit nicht Zeit? (Does time not have time?)

Friedrich Nietzsche

Beyond Good and Evil

Loosely related to unitarity, but more general, is the concept of determinism. This is usually weakened in
quantum mechanics anyway since in general one makes only probabilistic statements. Nevertheless, the wave
function is determined at all times by its initial values, which is sometimes seen as the appropriate
substitute for deterministic behavior. In loop quantum cosmology the situation again changes slightly since,
as discussed in Section 5.19, the wave function may not be determined by the evolution equation
everywhere, i.e., not at points of classical singularities, and instead may acquire new conditions on its initial
values. This could be seen as a form of indeterministic behavior, even though the values of a wave
function at classical singularities would not have any effect on the behavior of non-degenerate
configurations.4
(If they had such an effect, the evolution would be singular.) In this situation one deals with determinism in
a background-independent context, which requires a new view.

In fact, rather than interpreting the freedom of choosing values at classical singularities as
indeterministic behavior, it seems more appropriate to see this as an example for deterministic behavior in a
background-independent theory. The internal time label first appears as a kinematical object through
the eigenvalues of the triad operator (49). It then plays a role in the constraint equation (53) when
formulated in the triad representation. Choosing internal time is just done for convenience, and it is the
constraint equation that must be used to see if this choice makes sense in order to formulate
evolution. This is indeed the case at non-zero , where we obtain a difference operator in the
evolution parameter. At zero , however, the operator changes and does not allow us to
determine the wave function there from previous values. Now, we can interpret this simply
as a consequence of the constraint equation rejecting the internal time value . The
background-independent evolution selects the values of internal time it needs in order to propagate a wave
function uniquely. As it turns out, is not always necessary for this and thus simply
decouples. In hindsight, one could already have split off from the kinematical Hilbert space,
thereby removing the classical singularity by hand. Since we did not do this, it is the evolution
equation that tells us that this is happening anyway. Recall, however, that this is only one possible
scenario obtained from a non-symmetric constraint. For the evolution (55) following from the
symmetric constraint, no decoupling happens and is just like any other internal time
value.